A simplistic view of a material's glass transition temperature (Tg) is the temperature below which molecules have very little mobility. On a larger scale, polymers are rigid and brittle below their glass transition temperature and elastic above it. Tg is usually applicable to amorphous phases and is commonly applicable to glasses and plastics.
A fuller discussion of the Tg requires an understanding of mechanical loss mechanisms (vibrational and resonance modes) of specific (and usually common in a given material) functional groups and molecular arrangements. Things like heat treatment and molecular re-arrangement, vacancies, induced strain and other factors affecting the condition of a material may have an effect on Tg ranging from the subtle to the dramatic. Tg is dependant on the viscoelastic materials properties, and so varies with rate of applied load (silly putty is a good example of this, as is stiff cornflour/water mixtures - pull slowly and they flow, pull rapidly and they shatter).
In polymers, Tg is often expressed as the temperature at which the gibbs free energy is such that the activation energy for the cooperative movement of 50 or so elements of the polymer is exceeded. This allows moecular chains to slide past each other when a force is applied. From this definition, we can see that the introduction of side chains and relatively stiff chemical groups (such as benzene rings) will interfer with the flowing process and hence increase Tg.
In glasses (including amorphous metals and gels), Tg is related to the energy required to break and re-form covalent bonds in a somewhat less than perfect (may be regarded as an understatement) 3D lattice of covalent bonds. The Tg is therefore influenced by the chemistry of the glass. Eg. add B, Na, K or Ca to a silica glass, which have a valency less than 4 and they help break up the 3D lattice and reduce the Tg. Add P which has a valency of 5 and it helps re-establish the 3D lattice, increasing Tg.
The Space Shuttle Challenger disaster was caused by a rubber O-ring that was below its glass transition temperature and thus could not flex adequately to form a proper seal around one of the two solid rocket boosters.